Quantum symmetry. (English) Zbl 0762.17013

Gerstenhaber, Murray; Giaquinto, Anthony; Schack, Samuel D.

Quantum groups, Proc. Workshops, Euler Int. Math. Inst. Leningrad/USSR 1990, Lect. Notes Math. 1510, 9-46 (1992).

Let \(G\) be an algebraic group. Then \(O(G)\) denotes the bialgebra of polynomial functions on \(G\). In the present paper the authors treat deformations of \(O(G)\) dealing with \(G\) being \(\mathrm{GL}(n)\) or \(\mathrm{SL}(n)\). The bialgebra cohomology shows that if \(G\) is a reductive Lie group then any deformation of \(O(G)\) is equivalent to a preferred deformation in which the coproduct is strictly unchanged. Thus, to describe all deformations of \(O(G)\) it is sufficient to find all deformed multiplications. Using the fact that the preferred deformation for quantum linear space is given explicitly by Giaquinto’s formula (due to the second author), it became possible to indicate a family of preferred deformations of \(O(\mathrm{GL}(n)\), and \(O(\mathrm{SL}(n))\). The authors conjecture that they in fact found all such deformations of \(O(\mathrm{GL}(n))\) and \(O(\mathrm{SL}(n))\).
[For the entire collection see Zbl 0741.00068.]

Reviewer: A. Stolin (Rehovot)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16T10	Bialgebras
16T20	Ring-theoretic aspects of quantum groups
16T25	Yang-Baxter equations

Keywords:

bialgebra deformation; quantum symmetric group; quantized enveloping algebra; Yang-Baxter equation; bialgebra cohomology; preferred deformation

Citations:

Zbl 0741.00068